Regular Regimes of the Three Body Harmonic System

TL;DR

This paper analyzes the regular and chaotic regimes of a symmetric three-mass harmonic system, revealing how energy levels influence the transition between order and chaos through analytical and numerical methods.

Contribution

It provides a comprehensive analytical characterization of the system's regular regimes using perturbative methods and KAM theory, linking these to observed chaotic behaviors.

Findings

Low-energy regime described by Birkhoff normal form

High-energy regime exhibits regularity with a dominant frequency

Transition between regimes governed by regular solution structures

Abstract

The symmetric harmonic three-mass system with finite rest lengths, despite its apparent simplicity, displays a wide array of interesting dynamics for different energy values. At low energy the system shows regular behavior that produces a deformation-induced rotation with a constant averaged angular velocity. As the energy is increased this behavior makes way to a chaotic regime with rotational behavior statistically resembling L\'evy walks and random walks. At high enough energies, where the rest lengths become negligible, the chaotic signature vanishes and the system returns to regularity, with a single dominant frequency. The transition to and from chaos, as well as the anomalous power law statistics measured for the angular displacement of the harmonic three mass system are largely governed by the structure of regular solutions of this mixed Hamiltonian system. Thus a deeper understating of the system's irregular behavior requires mapping out its regular solutions. In this work we provide a comprehensive analysis of the system's regular regimes of motion, using perturbative methods to derive analytical expressions of the system as almost-integrable in its low- and high-energy extremes. The compatibility of this description with the full system is shown numerically. In the low-energy regime, the Birkhoff normal form method is utilized to circumvent the low-order 1:1 resonance of the system, and the conditions for Kolmogorov-Arnold-Moser theory are shown to hold. The integrable approximations provide the back-bone structure around which the behavior of the full non-linear system is organized, and provide a pathway to understanding the origin of the power-law statistics measured in the system.